Frequentist uncertainties on neural density ratios with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>w</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mtext> </mml:mtext><mml:msub><mml:mi>f</mml:mi><mml:mi>i</

We introduce <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>w</a:mi><a:mi>i</a:mi></a:msub><a:msub><a:mi>f</a:mi><a:mi>i</a:mi></a:msub></a:math> ensembles as a novel framework to obtain asymptotic frequentist uncertainties on density ratios, with a particular focus on neural ratio estimation in the context of high-energy physics. When the density ratio of interest is a likelihood ratio conditioned on parameters, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>w</c:mi><c:mi>i</c:mi></c:msub><c:msub><c:mi>f</c:mi><c:mi>i</c:mi></c:msub></c:math> ensembles can be used to perform simulation-based inference on those parameters. After training the basis functions <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>f</e:mi><e:mi>i</e:mi></e:msub><e:mo stretchy="false">(</e:mo><e:mi>x</e:mi><e:mo stretchy="false">)</e:mo></e:math>, uncertainties on the weights <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>w</i:mi><i:mi>i</i:mi></i:msub></i:math> can be straightforwardly propagated to the estimated parameters without requiring extraneous bootstraps. To demonstrate this approach, we present an application in quantum chromodynamics at the Large Hadron Collider, using <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>w</k:mi><k:mi>i</k:mi></k:msub><k:msub><k:mi>f</k:mi><k:mi>i</k:mi></k:msub></k:math> ensembles to estimate the likelihood ratio between generated quark and gluon jets. We use this learned likelihood ratio to estimate the quark fraction in a synthetic mixed quark/gluon sample, showing that the resultant uncertainties empirically satisfy the desired coverage properties.